Angle, area and isometries in Finsler spaces (Q2866251)

The author defines a Minkowski (Landslerg) measure of two rays \(a,b\in\mathrm{T}_{p_0}\) in the tangent space to a Finsler manifold \(M\) in \(p_0\in M\), out of the origin. He proves that this measure is additive, symmetric and it reduces to the Euclidian measure for the Euclidian spaces. Moreover, he shows that the Landsberg measure of every straight line is \(\pm\pi\) if and only if the Finsler metric is homogeneous, or equivalently if the indicatrix is symmetric. Then, considering two Finsler spaces \(F^n=(M,\mathcal F)\) and \(\overline{M}^n=(\overline{M},\overline{\mathcal F})\), he proves that a diffeomorphism \(\varphi:M\mapsto\overline{M}\) is a Finslerian isometry iff it preserves angles (and the 2-dimensional area). Next he gives another necessary and sufficient condition for \(\varphi\) to be a Finslerian isometry and shows that its verification reduces to the investigation of the isometries of two Riemannian spaces over \(M\) and \(\overline{M}\) resp., one being arbitrary.